Special issue on Selected papers from the Thirtieth International Symposium on Graph Drawing and Network Visualization, GD 2022
Empty Triangles in Generalized Twisted Drawings of $K_n$
Vol. 27, no. 8, pp. 721-735, 2023. Regular paper.
Abstract Simple drawings are drawings of graphs in the plane such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once, and there is a ray emanating from $O$ which crosses every edge exactly once. We show that all generalized twisted drawings of $K_n$ contain exactly $2n-4$ empty triangles, by this making a substantial step towards proving the conjecture that any simple drawing of $K_n$ contains at least $2n-4$ empty triangles.

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Submitted: October 2022.
Reviewed: January 2023.
Revised: February 2023.
Accepted: April 2023.
Final: April 2023.
Published: November 2023.
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